#### A femtoscopic correlation analysis tool using the Schrödinger equation (CATS)

Eur. Phys. J. C
A femtoscopic correlation analysis tool using the Schrödinger equation (CATS)
D. L. Mihaylov 1
V. Mantovani Sarti 1
O. W. Arnold 1
L. Fabbietti 0 1
B. Hohlweger 1
A. M. Mathis 1
0 Excellence Cluster Origin and Structure of the Universe , Boltzmannstr. 2, 85748 Garching , Germany
1 Physik Department E62, Technische Universität München , James-Franck-Str. 1, 85748 Garching , Germany
We present a new analysis framework called “Correlation Analysis Tool using the Schrödinger equation” (CATS) which computes the two-particle femtoscopy correlation function C (k), with k being the relative momentum for the particle pair. Any local interaction potential and emission source function can be used as an input and the wave function is evaluated exactly. In this paper we present a study on the sensitivity of C (k) to the interaction potential for different particle pairs: p-p, p- , K−-p, K+-p, p- − and - . For the p-p Argonne v18 and Reid Soft-Core potentials have been tested. For the other pair systems we present results based on strong potentials obtained from effective Lagrangians such as χ EFT for p- , Jülich models for K(K¯ )-N and Nijmegen models for - . For the p- − pairs we employ the latest lattice results from the HAL QCD collaboration. Our detailed study of different interacting particle pairs as a function of the source size and different potentials shows that femtoscopic measurements can be exploited in order to constrain the final state interactions among hadrons. In particular, small collision systems of the order of 1 fm, as produced in pp collisions at the LHC, seem to provide a suitable environment for quantitative studies of this kind.
1 Introduction
Femtoscopy has been mainly employed so far to study the
properties of the particle emitting source in heavy-ion
collisions by analyzing particle pairs with low relative momentum
undergoing a known interaction [
1, 2
]. In heavy-ion
collisions pion femtoscopy has been the most common tool to
get insight into the time-space evolution of the produced
medium [
3–13
].
Unlike heavy-ion collisions, nucleon–nucleon (NN)
collisions are not affected by the formation of a medium such as
the Quark–Gluon–Plasma. This means that the time
evolution of the source can be neglected and the hadron interaction
is not influenced by collective processes.
More generally, femtoscopy allows to describe any
hadronhadron correlation involving both (anti)mesons and (anti)
baryons. Recent femtoscopy studies in pp at √s = 7 TeV
and p–Pb at √sNN = 5.02 TeV showed that in these
systems a smaller source size radius is extracted compared to
heavy-ion collisions [
14–16
]. When mesons are considered
the correlation function is affected by a mini-jet background
over the whole k range [
14, 15
]. This effect is also visible in
baryon-antibaryon correlations (B–B¯ ), where partons from
the colliding protons hadronize in particle cones containing
a B–B¯ pair. The production of such pairs is enhanced due
to the intrinsic baryon number conservation and therefore
shows a strong kinematic correlation which overlaps to the
true femtoscopic signal in the final correlation function. Since
these mini-jets are not present in the correlation function of
B–B/B¯ –B¯ [17], those systems are well suited to study the
final state interaction.
The femtoscopic formalism includes the computation of
the correlation function for a given source function and
interaction potential [
2
]
C (k) =
S(r) | ψk (r) |2 d3r,
(1)
where k = (| p1 − p2 |)/2 is the reduced relative momentum
in the center of mass of the pair (p1 +p2 = 0), r is the relative
distance between the two particles, S(r) is the source function
and ψk (r) is the two-particle wave function.
The above formula is based on the assumptions that space
and momentum emission coordinates are uncorrelated and
the emission process is time independent [
2
]. The dynamics
of the processes is explicitly embedded on one hand in the
emitting source S(r), which depends on the colliding system,
and on the other hand in the interaction potential between the
two particles expressed by the relative wave function ψk (r).
The source could be either parametrized using a specific
distribution function, e.g. a Gaussian or a Cauchy
function, or extracted directly from transport models such as
EPOS [
18
]. The pair wave function can be obtained by
solving the Schrödinger equation for a specific interaction
potential and a popular tool to do that is the Correlation Afterburner
(CRAB) [
19
]. This afterburner only delivers the asymptotic
solution at large relative distances which eventually leads to
a wrong correlation function at small distances. For
heavyion collisions the source radius is typically between 2 and
6 fm [
3–7
]. However in pp collisions at LHC energies the
extracted source can be even smaller than 1.5 fm [
14,15
], and
the EPOS transport model predicts a non-Gaussian source
which peaks at distances of around 1 fm (see Fig. 1).
This calls for a tool capable of providing an exact solution
of the Schrödinger equation valid also at short distances.
Another common approach based on solving the
Schrödinger equation and used to study p–p correlations is the
Koonin model [
20
]. Here a Gaussian source distribution
is assumed and the relative wave function is obtained by
employing a Coulomb potential and a Reid Soft-Core
potential for the strong interaction [
21
].
Analytical solutions to compute C (k) do exist, but they
are limited by different approximations. One of the most
popular analytical models was developed by Lednický and
Lyuboshits [
22
]. The emission source is assumed to have a
Gaussian shape and the wave function is modeled within the
effective range approximation, using the scattering length
(a0) and the effective range of the potential (re f f ). Such a
parametrization is only sensitive to the asymptotic region of
the interaction. Due to its long range nature, the Coulomb
potential is normally not considered in the Lednický model
and can at most be treated in an approximate way [
13
].
All the above-mentioned issues motivated the
development of a new femtoscopy tool called “Correlation Analysis
Tool using the Schrödinger equation” (CATS).
CATS is capable of evaluating numerically the full wave
function without using any approximations and computing
C (k) for different source functions and interaction potentials.
CATS is faster than most of the conventional numerical tools
and also more flexible since it can be used together with an
external fitter.
The main purpose of this work is to test the sensitivity of
the correlation function to different potentials and provide
qualitative examples.
In particular for p–p and p– interactions we only
investigate the most common potentials used in femtoscopy so far
and compare them to more up-to-date potentials.
The p–p correlation function has been studied by
comparing the Reid Soft-Core [
21
] potential adopted in most
femtoscopic analyses to the most recent Argonne v18 [
23
]
potential which provides a better description of scattering
data.
Similarly, for the p– combination we compare the
commonly used Usmani potential [
24
] to the most recent NLO
potential obtained in χ EFT calculations [
25
] which
significantly improves the agreement with available data.
For the K±–p interaction we employ the Jülich model [
26,
27
] and we investigate the interplay between the Coulomb
and the strong interaction in the correlation function.
At the moment, the limited amount of experimental data
for the – interaction do not allow to discriminate among
different potentials. In this paper we show that future
femtoscopic analyses for the – correlation function performed
with CATS open the possibility to differentiate between
different types of potentials.
For this pair, following the results presented in [
28
], we
adopt four potentials: two versions of hard-core Nijmegen
models (ND56 [
29
], NF50 [
30
]), a quark-model potential
with meson exchange effect included (fss2 [
31
]) and another
hard-core Nijmegen potential NF42 [
30
] which allows for a
bound state.
The first three ones have been chosen since they deliver
the best agreement with the recent – STAR
measurements [
12
] while the latter has been selected since it is
connected to a possible bound state and it shows a different
behavior as a function of the relative distance r between the
two particles.
For p– pairs we adopt the most recent potential obtained
by the HAL QCD lattice collaboration [
32,33
].
The most modern potentials can also be implemented in
CATS, however this would require their transformation to
a local form, nevertheless this is beyond the scope of the
current study.
This paper is organized as follows. In Sect. 2.1 we
introduce the formalism of the Schrödinger equation as embedded
in the algorithm, while in Sect. 2.2 we introduce the
interaction potentials used for the later investigations of the p–p,
p– , K(K¯ )–p, p– − and – systems.
In Sect. 3 we show our results for the correlation
functions of the above-mentioned pairs using different interaction
potentials and source sizes. A comparison with the emitting
source obtained from the EPOS transport model is presented
for p–p and p– pairs. In addition we show results for p–
− and – correlation functions including experimental
effects such as momentum resolution and feed-down decays.
Moreover we present a comparison between the extracted
source radius for p–p pairs obtained from the HADES
collaboration in p–Nb reactions at √sNN = 3.18 GeV and a
CATS fit of the same data.
Finally in Sect. 4 we present conclusions and future
outlooks. In addition in Appendix A we provide technical
information related to the installation of CATS and the numerical
methods used in the code.
2 CATS
2.1 Overview
CATS is designed to compute the two-particle correlation
function for any emission distribution and interaction
potential. This is achieved by numerically solving the Schrödinger
equation (SE) and by evaluating the convolution of the
resulting wave function with a source distribution (see Eq. (1)).
The two-particle stationary SE reads
− 2h¯μ2 ∇2ψ + V ψ = E ψ,
m1m2 is the reduced mass of the system.
where μ = m1+m2
By assuming a central interaction potential V , the total
wave function is separated into a radial term and an angular
term given by the spherical harmonics
ψk (r, θ , φ) = Rk (r )Ylm (θ , φ).
In particular the radial equation to be solved is
h2 d2u
¯
− 2μ dr 2 +
h¯2 l(l + 1)
V (r ) + 2μ r 2
u = E u,
where u(r ) = r R(r ).
(2)
(3)
(4)
lmax
l=0
lmax
l=0
The overall interaction potential V (r ) is the sum of a short
range strong contribution and a long range Coulomb
potential. Since we are looking for scattering states as a function
of the relative momentum k and not for bound states, the
energy E of the system needs to be fixed by using the
rela2 2
tion E = h¯2 μk . The scattered state will then approach the
asymptotic solution outside the range of the strong potential
leading to a free or a pure Coulomb wave function depending
on the charge of the particles involved.
In order to properly match the exact solution to the
asymptotic form we have to evaluate the phase shifts of the
corresponding wave function. For this purpose the SE is solved
by expanding the total wave function in partial waves
ψk (r, θ , φ) =
Rk,l (r )Ylm (θ , φ)
il (2l + 1) uk,l (r ) Pl (cos θ ).
r
(5)
where Pl (cos θ ) are the Legendre polynomials.
The sum over the partial waves runs until the convergence
of the solution is reached, and from there we obtain the total
wave function ψk (r) to be used in Eq. (1).
Taking into account the Fermi statistics of the particle pairs
and fixing the isospin (I ) configuration, only specific states
2S+1 L J are allowed depending on the particle species, where
we use the standard spectroscopic notation with S denoting
the total spin, L the orbital angular momentum and J the
total angular momentum of the pair. Moreover, since NN,
nucleon–hyperon (NY), kaon–nucleon (KN) and hyperon–
hyperon (YY) potentials might involve spin-orbit coupling
L¯ · S¯, the total angular momentum J¯ = L¯ + S¯ has to be taken
as a good quantum number to characterize the eigenstates
and has to be accounted for in the total degeneracy of the
states.
The total correlation function that CATS provides as an
output is then given by
C (k) =
w(I,S,L,J )C(I,S,L,J ),
(6)
states(I,S,L,J )
where each C(I,S,L,J ) is evaluated by means of Eq. (1) and
weighted by w(I,S,L,J ).
In experimental conditions with unpolarized particles we
should take into account the degeneracy in S, as well as in I .
Moreover for L > 0 states also the degeneracy in J has to
be considered. How to compute the corresponding weights
w(I,S,L,J ) is explained in Appendix A.
In CATS the source function is characterized in two ways.
One possibility is to define an analytical function which
models the emission source. It is typically based on a Gaussian
-20
0
1S0
3P0
distribution of the x ,y and z single particle coordinates. The
emission source S(r, θ , φ) is defined as the probability
density function (pdf) to emit a particle pair at a certain relative
distance r and relative polar and azimuthal angles θ and φ.
For an uncorrelated particle emission the Gaussian source
reads
(7)
(8)
1
S(r ) = (4πr02)3/2 exp
where r0 is the size of the source. Since no angular
dependence is involved, to obtain the probability of emitting two
particles at a distance r a trivial integration over the solid
angle is necessary
4πr 2
S4π (r ) = 4πr 2 S(r ) = (4πr02)3/2 exp
Another possibility is to sample the relative distance between
the particle pairs directly from the output of an event
generator such as EPOS.
In Fig. 1 we compare the shapes of differently sized
Gaussian sources to the EPOS source for p–p pairs obtained in a
simulation of pp collisions at 7 TeV. Notably the EPOS source
predicts that most of the particle pairs will be emitted at
distances below 2 fm, which is the typical range of the strong
interaction. Hence the asymptotic solution of the wave
function will no longer be valid in that region, highlighting the
necessity of an exact treatment of the problem. Moreover we
see that EPOS predicts a non-Gaussian source.
2.2 NN and NY interaction potentials
The NN interaction has been widely constrained thanks
to a large amount of scattering data along with data on
bound states such as the deuteron [
34
]. In the common
Yukawa picture the NN potential is described by means of
model-independent components as the long range
One-PionExchange term (OPE) and the π π exchange term for the
intermediate attraction [
35
]. The vector meson (typically ω)
exchange accounts for the repulsive core.
This repulsive contribution can be phenomenologically
modeled with a Woods–Saxon function.
All these features are included in the most common NN
potentials such as Bonn [
36,37
], Urbana [38], Paris [
39
],
Nijmegen [
40
], Reid [
21
] and Argonne [
23
] potentials. In
addition, the results obtained within a chiral effective field
theory approach (χ EFT) [
41–45
] and within the latest version
of the Nijmegen model ESC08 [
46
] have to be mentioned,
since they provide similar results for the scattering
parameters and both models show a nice description of the available
data.
Fig. 2 Phase shifts as a function of the relative momentum k for the
p–p system using the Argonne v18 potential (blue solid line) and the
Reid SC potential as used in CRAB calculations (green dashed line).
The phase shifts are taken from [
34
]
For the study of the p–p correlation function in this paper
we employ the Argonne v18 [
23
] and the Reid Soft-Core
(RSC) [
21
] potentials. The latter is the default choice in
femtoscopic analyses [
19,20
]. In the Koonin model only the RSC
singlet state 1 S0 is considered, while in the CRAB
calculations also the triplet 3 P2 state is included but without
spinorbit or tensor terms. Nevertheless, these terms are crucial
in order to reproduce the 3 P2 phase shift, as can be seen in
Fig. 2. Here the Argonne v18, which includes all s- and
pwave states and both the spin-orbit and tensor terms, results
in a much better agreement with the partial wave data in the
momentum range we are interested in.
It should be noted that a more recent version of the RSC
potential [
47
] is available. Nevertheless it is beyond the
purpose of the current work to study in details the subtle
differences between a large variety of NN potentials, since we do
not expect a large discrepancy in the p–p correlation function.
The range of relative momenta k relevant for femtoscopic
studies stops at kmax ≈ 200 MeV/c and the dominant
contribution in the phase shifts in this regime is mainly coming
from s- and p-wave states [
34
]. For this reason at the moment
we are neglecting the coupled channel to the 3 F2 wave.
Unfortunately in the nucleon–hyperon sector only 36
scattering data points have been measured [
50–55
] and the
available phenomenological potentials are fitted in order to
reproduce these data along with the accessible hypernuclei binding
energies. In particular, for the p– interaction we adopt the
potential introduced by Bodmer, Usmani, and Carlson [24],
which reproduces the available cross section data as shown
in Fig. 3. This potential is similar to a NN Argonne
interaction since it includes a repulsive Woods–Saxon core and
a two-pion intermediate exchange term. At the moment we
just consider s-waves, which account for both the singlet 1 S0
and the triplet 3 S1 state.
80
k (MeV)
To test the sensitivity of the correlation function to the p–
interaction, we have also used the results obtained by
nextto-leading order (NLO) χ EFT calculations [
25
]. In this work
the authors show that the agreement with the available data
in the S = − 1 sector is considerably improved with respect
to previous leading order results. A local potential for this
model is not yet available, nevertheless we were provided
with the radial wave function and used CATS to compute
the resulting correlation functions for different sources. The
scattering parameters of the two potentials used to model the
p– interaction are listed in Table 1.
Concerning the S = − 2 sector the interaction of
hyperons with nuclei is not well constrained experimentally or
phenomenologically [
56
]. Recently a -hypernucleus candidate
has been detected [
57
] and ongoing measurements suggest
that the N– interaction is weakly attractive [
58
].
Recent lattice QCD calculations from the HAL QCD
collaboration showed preliminary results on the p– −
correlation function based on local potentials in the s-wave singlet
and triplet states, both in I = 0 and I = 1 channels [
32, 33
].
The interaction potentials have been obtained in (2 +
1)flavor lattice QCD simulations close to the physical masses
of pions and kaons. In particular in the isoscalar 1 S0 channel
it has been shown that the p– − interaction is deeply
attracData
Usmani
NLO
Ratio to Usmani
Ratio to NLO
100
120
140
tive at intermediate distances and relatively weakly repulsive
at shorter distances. The isoscalar spin triplet state 3 S1 has a
shallow attractive well and a very repulsive core.
The I = 1 channel does not present any attraction in the
spin singlet state while a mildly attractive region enclosing a
repulsive inner core has been found in the spin triplet state.
These contributions might not be relevant for large source
sizes [32] but could impact the overall correlation function for
smaller emitting sources as obtained in elementary collisions.
2.3 KN and K¯ N interaction
As for the hyperon–nucleon case, the (anti)kaon–nucleon
system is a powerful tool to study the short range strong
interaction with the exchange of strangeness as a new degree
of freedom.
In particular kaons are suited to study the inner part of
nuclei since at low energies they interact rather weakly with
the nucleons [
56
]. On the contrary antikaons scattering on
nuclei is characterized by absorption processes that can lead
to the production of hyperons such as and [
56, 59
].
The understanding of the short range kaon–nucleon
interaction, already in vacuum, is fundamental to unveil more
complicated systems such as quasi-bound states near
threshold, kaonic atoms and kaonic clusters [
60–66
].
From a theoretical point of view, based on the
successful description of the NN interaction, several meson-baryon
exchange models, such as the Jülich model [
26, 27
], have
been applied so far in the description of the available KN
data. In this approach the short range repulsion in K+-N is
not well reproduced by the simple presence of the ω meson.
In order to reproduce the s-wave scattering parameters an
additional repulsive contribution is needed. The inclusion of
spin-dependent terms, such as one gluon exchange (OGE)
or exchange of the scalar-isovector meson a(980), improves
the agreement in the s-channel [67].
It is also worth mentioning that different theoretical
approaches based on quark models [
68–70
] and χ EFT
interactions [
71
] have also been used to investigate the kaon–
nucleon interaction.
In this work we adopted the above mentioned Jülich
model, which besides single particle exchange terms also
contains box diagrams allowing for intermediate states such
as N, , K(K¯ ), K∗(K¯∗). A local potential was not available
for this model, nevertheless we were provided with the exact
solution of the radial wave function, which was included in
CATS for the computation of the correlation function.
Besides the wave function corresponding to the strong
K(K¯ )–N interaction potential, we also take into account the
Coulomb interaction in an approximate way by multiplying
the (strong) wave function with the Gamow factor Ac(η) [
13
]
ψKN,K¯ N(r, θ , φ) = Ac(η)ψ s (r, θ , φ)
where η = 21 as kKN,K¯ N −1 and as = (mred z1z2e2)−1, with
z1,2 being the charge numbers of the two particles. A more
elaborate approximation based on the renormalization of the
asymptotic behaviour of the strong wave function has been
developed in the past [
72
] and will be used in future analyses.
2.4 YY interaction
In the YY sector the available experimental data on the
binding energy of hypernuclei, despite recent data on
6 He [
73,74
], do not allow to set tight constraints on the
nature of the underlying interaction [56].
Recently the STAR collaboration employed the
femtoscopy technique to study – correlations in Au-Au
collisions at √sNN = 200 GeV [
12
]. The reported shallow
repulsive interaction in this work is still under debate, since in an
alternative analysis, that considered the contribution of the
residuals in a more sophisticated way, a shallow attractive
interaction was confirmed [
28
]. One of the goals of this work
is to show that a detailed study of the – correlation
function might be a sensitive tool to extract information on the
interaction of the two particles.
For this purpose, we select four different potentials: three
meson-exchange Nijmegen models with a short range
hardcore (ND56 [
29
], NF50 and NF42 [
30
]) and one quark-model
potential which includes meson exchange effects as well
(fss2 [
31
]). The models ND56, NF50 and fss2 deliver the
best agreement with the – correlation data measured by
STAR. The NF42 has been chosen to show the sensitivity of
the correlation function to the presence of a possible bound
state, which is allowed by this potential.
The above mentioned potentials have been translated in a
local form by using a two-range Gaussian parametrization,
following the approach used in [
28
]
V
(r ) = v1e−r2/μ21 + v2e−r2/μ22 .
(10)
The parameters are fixed to values that reproduce the
scattering parameters obtained within each model. They are
sumμ2 (fm)
0.45
0.45
0.45
0.41
a0
marized in Table 2. In Fig. 4 the potentials are plotted as a
function of the relative distance r .
It is worthwhile mentioning that only the NF42 model
provides a positive scattering length a0 ≈ 3.7 fm due to the
presence of a bound state, while all the remaining potentials
lead to negative (attractive) values of a0 and effective ranges
between 4 and 5 fm.
3 Results
In Fig. 5 a comparison between the correlation functions
obtained by CRAB and CATS for p–p pairs is shown. In
particular two different transport models are used to model
the source: UrQMD [
75
] for simulating p–Nb reactions at
√sNN = 3.18 GeV and EPOS [
18
] for simulating pp
reactions at √s = 7 TeV. The p–Nb system is expected to have
a Gaussian source of around 2 fm [
76
], while the pp system
produces a much smaller source, comparable to a Gaussian
source of below 1 fm (see Fig. 1). As already pointed out the
CRAB tool provides only an approximate solution at small
distances. Figure 5 shows that in the case of the larger source
(the p–Nb system) the two models are in agreement within
few percent. However for pp collisions at 7 TeV the source is
much smaller and consequently a much larger discrepancy, of
up to 20%, is observed between CRAB and the exact CATS
solution.
2.5
3
2
1
)
k
(pp 1.5
C
0.5
1.1
itaoR0.91
0.8
0
Fig. 5 Correlation function for p–p pairs obtained within UrQMD
simulations for p–Nb reaction at √sNN = 3.18 GeV (left panel) and within
EPOS for pp collisions at √s = 7 TeV. For both setups results are
20
40
60
100
120
140
160
20
40
60
100
120
140
160
presented for CATS calculations with Av18 and CRAB with a modified
Reid SC potential (see 2.2 for details)
EPOS
CATS Av18
CRAB Reid SC
1.2
1
0.8
In Fig. 6 we show the correlation function C (k) obtained
by CATS for p–p pairs (upper panel) and p– pairs (lower
panel) with different interaction potentials and Gaussian
source sizes.
In particular for p–p pairs we perform the calculations for
the RSC potential as used in the Koonin and CRAB models
and compare them to the Av18 potential. The profile of the
resulting correlation functions reproduce the main features
coming from the interplay between the attractive part of the
NN potential and the short-range repulsive core along with
contributions from Coulomb and Fermi-Dirac statistics. The
repulsive interaction is visible between 60 < k < 200 MeV
since C (k) is below unity. This effect is enhanced for smaller
sources.
The strength of the p–p correlation signal increases by
almost a factor 2.5 going from r0 = 2 fm to r0 = 0.85 fm.
The differences among the potentials are negligible for most
of the source sizes. Nevertheless it is evident that for sources
below 1 fm deviations up to 10% are present at k > 150 MeV.
The p– pair is neither affected by Coulomb
interaction nor by quantum statistics. As a result the shape of
the corresponding correlation function C (k) is only affected
by the strong interaction potential. Here results are shown
for the Usmani potential and the χ EFT NLO potential (see
Fig. 7 Correlation functions
for p–p pairs (left) with Av18
potential and p– pairs (right)
with χ EFT NLO potential
(see 2.2). The calculations are
performed by assuming
Gaussian emitting sources of
radii r0 = 1.2 fm (solid blue
line), r0 = 0.85 fm (dashed
green line) and by using EPOS
transport model (red dot-dashed
line)
6
5
4
2
1
r 0=1.2 fm
r0=0.85 fm
EPOS
0
50
150
200
Sect. 2.2). Both of these potentials possess a
phenomenological repulsive core which is dominated by an intermediate
attraction as k decreases. This results in a rise of C (k) at lower
momenta. The two potentials produce very similar
correlation functions and the differences are mostly associated with
the slightly more attractive 3 S1 state in the Usmani potential.
This can be easily understood by comparing the scattering
parameters (see Table 1) of the Usmani and the chiral
potential. While the scattering lengths in the singlet state are almost
identical, they do differ in the triplet state, where the Usmani
potential is slightly more attractive, resulting in an enhanced
correlation.
In addition for the NLO correlation function we compare
the exact solution to the approximate Lednický model. As
expected we observe a significant deviation between the exact
and approximate solution at small source radii. The Lednický
model produces an enhanced correlation signal because it
contains the NLO scattering parameters that account only
for the average attractive interaction. On the contrary the
correlation function obtained feeding the exact NLO wave
function in CATS shows the repulsive component expected
within this model.
It is clear that for sources ≥ 2 fm the correlation function
is not sensitive to the repulsive core both for p–p and p–
pairs. However, in the case of smaller sources, as shown in
the last panel of Fig. 6, the correlation functions split up and
remain different even for relative momenta k > 150 MeV.
This region is commonly used in femtoscopy analyses as a
correlation free baseline to which the overall normalization
of C (k) is applied. Due to to the possible non-flat correlation
in that region particular care has to be considered in including
the correct potential in the correlation function.
In Fig. 7 the emitting source has been directly taken from
the EPOS model simulating pp collisions at √s = 7 TeV.
In this transport code the evaluated source peaks at very low
distances and has a long tail, as already shown in Fig. 1.
Moreover the EPOS source presents a non-trivial angular
dependence of the emission source. The sources for both the p–p
and p– systems in EPOS are the same. However, if a
Gaussian parametrization is employed to reproduce the correlation
functions obtained with EPOS, different source sizes r0 for
both systems are needed. This suggests that when
investigating experimental baryon-baryon correlations in small
systems, particular care has to be taken in the assumption of the
profile of the emitting source.
Results for K−–p and K+–p correlations for different
source sizes are shown in the upper and lower panel of Fig. 8.
The correlation function including only the Coulomb
interaction is plotted in order to see the effect of the strong potential.
The K−–p correlation function is deeply affected by the
strong interaction already at k ≈ 200 MeV and it dominates
the low k region. As expected both these effects are enhanced
when the source gets smaller.
The opposite scenario is depicted for the K+–p pair,
where the Coulomb and the strong repulsion lead to an
anticorrelated signal at low momenta.
In Fig. 9 we show the expected theoretical p– −
correlation function. We compare C (k) for a pure Coulomb
interaction with the correlation function including the
preliminary strong potential from the HAL QCD collaboration. The
included channels are the (I = 0, S = 0), (I = 0, S = 1),
(I = 1, S = 0) and (I = 1, S = 1), where only s-waves are
considered. There is a clear enhancement of the correlation
signal at small relative momenta k, which is a result of the
attractive I = 0 channel. This effect is further increased for
small emitting sources.
In Fig. 10 we present results for – analysis with several
interaction potentials (see Sect. 2.4). As a comparison the
correlation function obtained only with the quantum statistics
included is presented. The latter is responsible for the limit
value C (k) = 0.5 as the relative momentum k tends to zero.
The correlation function shows a strong sensitivity to
the interaction potential and moreover the shape of C (k)
is deeply affected by the source size. This behaviour can
be explained easily by looking at the potentials shown in
Fig. 4 and their corresponding scattering parameters listed in
Table 2. The most binding potentials, NF42 and ND56, lead
to a correlation function that is significantly enhanced for
smaller sources as it becomes sensitive to the deep
attractive contribution. The weakly attractive potentials, NF50
and fss2, also show an enhancement compared to the
quantum statistics baseline, nevertheless the correlation signal
is much weaker due to the presence of a large repulsive
core.
When making predictions about the correlation function
it is important to take into account the expected experimental
limitations. In particular the momentum resolution and the
residual correlations contributing to the femtoscopic signal
can distort the correlation function [
77
]. To estimate how the
p– − and – correlation functions would be affected by
100
k (MeV)
150
50
100
150
k (MeV)
Strong+Coulomb
Coulomb
Fig. 11 Correlation function
for p– − (top) and –
(bottom) with feed-down
coefficients and moment
resolution included. For p– −
pair, the preliminary HAL QCD
potential [
32,33
] with both
I = 0 and I = 1 contributions
has been employed (see 2.2).
Different emitting source sizes
are considered
4
) 3
k
(
_ Ξ
p p
x
e C 2
r 0=0.85 fm
Strong+Coulomb
Coulomb
50
100
k (MeV)
150
100
k (MeV)
150
100
k (MeV)
150
experimental corrections, we use the linear decomposition
of the correlation function as described in [
77
]. In short one
assumes that only a fraction λ of the particle pairs represent
the actual correlation of interest and the rest of the pairs are
either misidentified or stemming from the decays of
resonances (feed-down). For the purpose of a qualitative
prediction we will assume that all feed-down contributions have
a flat correlation distribution. In that case the experimental
correlation function Cex p(k) is expressed by
Cex p(k) − 1 = λ(C (k) − 1).
(11)
In Fig. 11 we plot the estimated experimental correlation
functions for p– − and – pairs. For both systems we
apply the momentum smearing matrix used for the p–
correlation in [
77
]. The λ coefficient for the – correlation is
taken from the same reference and has the value of 32%. To
obtain a reasonable λ coefficient for p– − we use the proton
feed-down and impurities as evaluated in [
77
], while for the
− particle we assume 90% purity and feed-down fractions
of 32% from (1530) [
78
] and ≈ 1% from −. The
corresponding λ coefficient for the genuine p– − correlation has
a value of 52%.
One can see in Fig. 11 that for the – case the effect
of the different potentials on the correlation function is
significantly damped by the inclusion of expected experimental
effects. This implies that very large statistics are needed to
reach a high precision in the determination of the
scattering parameters or in testing different models. For the p– −
case the effect of the strong potential is clearly
distinguishNF42
NF50
ND56
fss2
w/o strong int.
100 150 200
k (MeV)
HADES data
CATS Av18
r HADES=2.02 ± 0.01 fm
r CATS=2.01 ± 0.01 fm
2.5
2
1
()K 1.5
p
p
C
able from the Coulomb-only interaction even if realistic λ
parameters are considered.
In order to apply the CATS framework to
experimental data we present, in Fig. 12, the result for the p–p
correlation function obtained in p–Nb collisions at √sNN =
3.18 GeV [
76
]. We evaluated C (k) in CATS using the Av18
potential with s- and p-waves included and performed a fit to
the data by multiplying C (k) with a normalization constant
N . The source is assumed to be Gaussian and the source
size is a free fit parameter. The HADES analysis extracts
a radius of r0 = 2.02 ± 0.01(stat) fm and our fit value is
2.01 ± 0.01(stat) fm, which is in perfect agreement with the
published data.
4 Conclusions and outlooks
In this paper we presented the new femtoscopy analysis
tool CATS, which allows for an exact computation of the
correlation function for different sources and potentials. To
obtain a solution valid for any source size CATS solves the
Schrödinger equation exactly, thus providing the possibility
to investigate not only heavy-ion reactions but small collision
systems as well.
In the present work we show the theoretical correlation
functions for the p–p, p– , K(K¯ )–p, p– − and – particle
pairs, by using different models for the interaction potential.
A comparison between CATS and CRAB reveals that
for an emitting source of around 2 fm the correlation
functions are quite similar. Nevertheless, for small colliding
systems, such as pp at TeV energies, the exact treatment of the
Schrödinger equation strongly deviates from the approximate
solution of CRAB. Thus CATS provides a unique opportunity
to investigate the short-range interaction between particles in
small systems.
We show that both the p–p and p– correlation functions
are affected by the repulsive core of their interaction
potentials, which is mostly evident for small emitting sources. This
effect is particularly relevant for momenta above 100 MeV,
where the exact solution deviates significantly compared to
approximate models like Lednický.
Moreover for the p–p and p– correlations we test
different emission sources by using the EPOS transport model. In
Fig. 1 we show that the emitting source provided by EPOS
is completely different from the traditionally Gaussian form
used so far in femtoscopic analyses. The results presented
here suggest the need for a more detailed analysis of the
profile of the emitting source beyond the traditional Gaussian
parametrization, whenever dealing with small systems.
The investigation of the K(K¯ )–p correlation function
shows a relevant modification due to the strong interaction.
This effect increases with a decreasing source size.
The p– − correlation function is calculated by
employing a preliminary local potential obtained from recent
lattice HAL QCD results. We observe that for small emitting
sources, as expected in pp collision systems, the strong
interaction modifies the correlation function significantly. This
implies that future experimental femtoscopic data on the p–
− correlation should be sensitive to this effect. This
statement is still valid if one includes momentum resolution
corrections and residual effects in the correlation function.
For the – correlation function we show results for
different interaction potentials, based on Nijmegen
bosonexchange models and quark models. Starting from source
sizes of r0 = 2 fm, the corresponding C (k) shows a
relevant sensitivity to the strong potentials. As the source radius
decreases the correlation function is modified by the
attractive contribution common to all of the above-mentioned
potentials. We also show that the inclusion of the expected
experimental effects makes the separation of the different
potentials extremely challenging and will require a lot of
statistics.
CATS was designed such that it can easily be used with an
external fitter. To demonstrate this we perform a refit of the p–
p correlation function measured by the HADES experiment
in p–Nb reactions at √sNN = 3.18 GeV. The obtained results
are in a perfect agreement with the published data.
The experimental database of femtoscopy is constantly
growing, thus in the near future it will be essential to
perform global analyses over all available data in order to get
the best possible constraints on the interaction potentials.
The flexibility of CATS allows it to be used as the base of
more complex frameworks and is therefore ready to fulfill
the requirements of the upcoming experimental analyses.
Acknowledgements The authors are grateful to Dr. Stefano Gandolfi,
Prof. Johann Haidenbauer, Prof. Tetsuo Hatsuda, Prof. Norbert Kaiser,
Prof. Kenji Morita and Prof. Wolfram Weise for the stimulating
discussions and the useful contributions to the work presented here. This
work is supported by the Deutsche Forschungsgemeinschaft through
Grant SFB 1258 “Neutrinos and Dark Matter in Astro- and Particle
Physics”. We also gratefully acknowledges the support by the grants
BMBF05P15WOFCA, DFG EClust 153, MLL, TU München
(Germany).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A
The “Correlation Analysis Tool using the Schrödinger
equation” is implemented in a C++ class called CATS,
alongside with few supporting classes that are provided in
the CATS package, and can be found here: http://www.
denseandstrange.ph.tum.de/index.php?id=78. In this section
section we would like to discuss some of the technical details
regarding the working principles of CATS. All information
refers to CATS version 2.7, any future changes will be marked
in the release notes.
The CATS software package contains basic examples
highlighting the main input parameters that the user needs
to set up in their code. Those include the basic properties of
the particle pair, e.g. reduced mass, charges etc. We would
like to focus on the two most important inputs, namely the
emission source and the interaction potential.
Appendix A.1: The source function
Commonly the source function is considered to have only
r dependence, r being the relative distance between the
two particles. However CATS uses a more generic
definition which allows an additional dependence on the relative
momentum k and the scattering angle θ = (r, k). The
specific implementation of the source and its parameters in
CATS allows to link it to an external framework to perform
data-fitting (see Fig. 12).
Alternately the source can be inserted into CATS from an
OSCAR 1997 A file [
79
]. This format is commonly used by
transport codes in relativistic heavy ion collisions and it
contains event-by-event information about the position and the
momentum four-vectors of all particles produced in the
simulation. Note that obtaining the source from simulations leads
to statistical uncertainties which result in a limited precision
when determining the source function. Thus CATS also
computes the uncertainty associated with the source function and
propagates it to the correlation function.
Appendix A.2: Computing the wave function
The heart of CATS is the numerical solver for computing
the wave function. The two body problem can be reduced
to a one-body problem simply by using the reduced mass
μ = (m1m2)/(m1 +m2) and the reduced relative momentum
k = 21 |p1 − p2|, where p1 and p2 are the momenta of the
two particles evaluated in their center of mass (CM) frame
of reference.
To obtain the SE solution we follow the standard
procedure of separation of variables and partial waves
decomposition (Eq. (A.1)) information on which, including the standard
notation, can be found in any quantum mechanics textbook
(e.g. [
80
])
ψk (r) = Rk (r )Y (θ ) =
Rk,l (r )Yl (θ )
=
lmax
il (2l + 1)Rk,l (r ) Pl (cosθ ).
lmax
l=0
l=0
For low-energy problems, as the one at hand, this series
converges quite fast. The Legendre polynomials Pl are
easily computed using the GNU scientific library (GSL) [
81
],
hence the only thing left to evaluate is the radial wave
function Rk,l (r ), which satisfies the radial Schrödinger equation
(RSE)
d2uk,l (r )
dr 2
=
2μVI,s,l, j (r )
h2
¯
+
l(l + 1)
r 2
− k2 uk,l (r ),
where uk,l (r ) = r Rk,l (r ) and VI,s,l, j (r ) is the interaction
potential between the particle species of interest. As marked
by the subindices the potential can depend on all quantum
numbers describing the pair and consequently the solutions
for the radial wave functions will differ between the
different states. How to combine the different solutions to obtain
the final correlation function will be explained at the end of
this subsection, but first let us turn the focus to solving the
Schrödinger equation for fixed values of k, I , s, l and j .
To avoid the overuse of indices, the radial wave functions
uk,l (r ), Rk,l (r ) and the potential VI,s,l, j (r ) will be written as
u(r ), R(r ) and V (r ) respectively. In order to solve Eq. (A.2)
numerically we employed a simple Euler-method equipped
with an adaptive grid which automatically adjusts its step
size when computing u(r ). The basic idea is to make the
step smaller when computing regions of the wave function
that have a strong non-linear behavior, i.e. a large second
derivative. The discretized version of Eq. (A.2) using the
Euler method and a dynamic step size reads
Δi
ui+1 = ui 1 + Δi−1
Δi 2
− ui−1 Δi−1 + Fi ui Δi ,
≈const
with
Fi =
2μV (r )
h2
¯
+
l(l + 1)
r 2
− k
2 ,
where ui is the value of u(r ) at the i -th point on the discrete
r -grid, Δi = ri+1 − ri is the distance on the grid to the next
evaluation point. The relation ui = Fi ur is given by
combining Eqs. (A.2) and (A.4), thus in order to adjust the step
size one could simply keep the last term in Eq. (A.3) constant.
This ensures that Δi2 ∝ 1/ui and the non-linear increase of
ui+1 is kept constant throughout the computation. We have
explored the possibility to select a more sophisticated
numerical method, e.g. a Numerov method. However the inclusion
of an adaptive grid tends to become quite complicated for
higher-order numerical solvers. We have found out that the
Euler method with an adaptive provides a faster solution,
without compromising accuracy or precision, compared to
the Numerov method without an adaptive grid. For this
reason we have decided to employ the former.
One additional problem that arises when solving the
Schrödinger equation is related to the boundary conditions.
In particular Eq. (A.3) can be used to get the full wave
function only if the first two points u0,1 are known. Note that
u(r ) = r R(r ), hence u(0) = u0 = 0 can be used as the
starting point of the solver. However the second point of the
wave function depends on the potential and is thus a priori
not known. Nevertheless if the second point u1 is chosen
randomly Eq. (A.3) will still yield the correct shape of the
wave function at the expense of a wrong absolute
normaliza(A.1)
(A.2)
(A.3)
(A.4)
Fig. 13 An example for the
CATS input when both the
sand p-wave interactions are
included. There are four distinct
interaction channels:
ψ0 = ψ1S0 + ψ1P1 + ...,
ψ1 = ψ3S1 + ψ3P0 + ...,
ψ2 = ψ3S1 + ψ3P1 + ... and
ψ3 = ψ3S1 + ψ3P2 + .... The
dots represent all partial waves
beyond the p-wave, which are
assumed to have no influence on
C(k) and are thus considered
irrelevant. The grayed out states
are not allowed in the case of
p–p correlation, but can be
included if needed, e.g. for p–
correlations
tion. To obtain the correct normalization a second boundary
condition is needed.
A straightforward approach to overcome this issue is to
use the asymptotic solution. For a short range interaction
potential V (r ), the evolution of the asymptotic wave function
is governed by a phase shifted free or Coulomb wave. CATS
is able to check when the solution has reached its asymptotic
region and then simply match the numerical solution to the
asymptotic one. This procedure allows to determine not only
the normalization of u(r ) but the phase shift of the potential
V (r ) as well (see Fig. 2). The total wave function ψk (r) is
given by the sum over all l partial waves
ψk (r) =
ψk,l (r) =
ψk,l (r) +
ψk,l (r) . (A.5)
ln
l=0
lmax
l=ln +1
lmax
l=0
numerical
asymptotic
Here the value of lmax is determined by the condition for
convergence, namely when l > lmax the partial waves are
practically zero in the asymptotic range. The sum is then split
in two parts – first all numerically evaluated partial waves are
summed up and than all remaining partial waves are added
using their asymptotic solution obtained from GSL.
Another important thing to consider is the possibility to
have different spin and isospin states. Moreover since the
total angular momentum J is a good quantum number for
potentials with spin-orbit terms, the degeneracy in J has to
be taken into account as well. All of these contributions will
result in different partial wave states 2S+1 L J and the user
should provide CATS with all relevant information in order
to get a correct total correlation function. The way this is
achieved is by introducing the notation of interaction
channels (see Fig. 13). An interaction channel is defined as the
total wave function for a specific spin and isospin state and
each partial wave that is included in the computation is
considered to be in a fixed J state. The user should carefully
Channel 0
Channel 1
Channel 2
Channel 3
Total spin 0
Weight 1/4
Total spin 1
Weight 3/4
examine all possible permutations to include each channel
that contributes to the final correlation and should provide
CATS with the probabilities that the particle pair can be found
in particular channel.
We provide an example for the p–p interaction. For this
system the isospin can only be 1, hence it is not considered
in the calculation. However the total spin S of the system can
be either 0 or 1. Since those states correspond to a singlet
and a triplet configuration respectively, as long as there is no
preferred polarization the probability to be in the singlet state
is 1/4 and in the triplet 3/4. Those weights can be computed
from the simple formula regarding spin degeneracy
(2S + 1)
w(S) = (2s1 + 1)(2s2 + 1)
,
(A.6)
where s1 and s2 are the spins of the individual particles. If
one would consider only the s-waves in the computation the
relevant partial waves are 1 S0 and 3 S1. In fact for p–p, due
to the quantum statistics of identical fermions, the 3 S1 state
is not allowed and therefore no potential will be considered
in CATS. Nevertheless it is very important to note that the
weights of 1/4 and 3/4 still need to be set, since even if
the 3 S1 partial wave is canceled out for p–p, the total wave
function of the S = 1 is still included in C (k) with a flat
contribution. The situation becomes a bit more complex if
the p-waves are to be considered as well. In the S = 0 the
1 P1 is the only p-partial wave and for p–p it is canceled out
again due to the quantum statistics. However the S = 1 state
has 3 possible p-waves, namely 3 P0, 3 P1 and 3 P2. These
states have the same S and L quantum numbers, but different
J . Hence to compute their relative contributions one needs
to consider the degeneracy in J which is given by
(2 J + 1)
w(J ) = (2L + 1)(2S + 1) .
(A.7)
The total weights will be given by:
w(S,L,J ) = w(S) · w(J ) =
(2S + 1)
(2 J + 1)
(2s1 + 1)(2s2 + 1) (2L + 1)(2S + 1)
.
(A.8)
(A.9)
In Fig. 13 we show a schematic representation of the above
example.
From the user’s perspective, the inputs that CATS has to
be provided with are the number of interaction channels,
the number of partial waves to be included in each channel
and finally the potentials for each partial wave state. In the
example from Fig. 13 the user needs to define 4 channels
each containing two partial waves. However, due to the Pauli
blocking, the 1 P1 and 3 S1 states can be safely left undefined
in the code.
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